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Finite-Range Repulsive Systems of Finitely Many Particles

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Abstract.

We consider the dynamics of finite systems of spherical particles where a fraction of the kinetic energy may be lost during collisions. We show that the set of initial configurations leading to infinitely many collisions in finite time can have positive measure, contrary to the hard-collision (energy conservation) case, in which this particular set is claimed to be empty. We also show that after sufficient time a system will decouple into maximal subsystems. This generalizes proofs that in the hard-sphere case there can be at most finitely many collisions for almost all initial configurations.

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References

  1. Alexander, R.K.: The infinite hard sphere system. Ph.D. thesis, University of California at Berkeley, 1975

  2. Benedetto, D., Emanuele, C.: The collapse phenomenon in one-dimensional inelastic point particle systems. Physica D 132, 457–475 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Springer-Verlag, New York, 1994

  4. Constantin, P., Grossman, E., Mungan, M.: Inelastic collisions of three particles on a line as a two-dimensional billiard. Physica D 83, 409–420 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Illner, R.: On the number of collisions in a hard sphere particle system in all space. Transport Theory and Stat. Phys. 18, 71–86 (1989)

    MathSciNet  MATH  Google Scholar 

  6. Illner, R.: Finiteness of the number of collisions in a hard sphere particle system in all space II: arbitrary diameters and masses. Transport Theory and Stat. Phys. 19, 573–579 (1990)

    MathSciNet  MATH  Google Scholar 

  7. Sandri, G., Sullivan, R.D., Noren, P.: Collisions of three hard spheres. Phys. Rve. Lett. 13, 743–745 (1964)

    Article  MATH  Google Scholar 

  8. Uchiyama, K.: On the Boltzmann-Grad limit for the Broadwell model of the Boltzmann equation. J. Stat. Phys. 52, 331 (1988)

    MathSciNet  MATH  Google Scholar 

  9. Vaserstein, L.N.: On systems of particles with finite–range and/or repulsive interaction. Commun. Math. Phys. 69, 31–56 (1979)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260

    Xinfu Chen

  2. Department of Mathematics and Statistics, University of Victoria, Victoria, B.C. V8W3P4, Canada

    Reinhard Illner

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  1. Xinfu Chen
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  2. Reinhard Illner
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Correspondence to Xinfu Chen.

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Communicated by D. Kinderlehrer

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Chen, X., Illner, R. Finite-Range Repulsive Systems of Finitely Many Particles. Arch. Rational Mech. Anal. 173, 1–24 (2004). https://doi.org/10.1007/s00205-004-0309-6

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  • DOI: https://doi.org/10.1007/s00205-004-0309-6

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